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...Contents singular value decomposition svd vectors best rank k approximations power method for computing the applications of principal component analysis clustering a mixture spherical gaussians an application to discrete optimization problem svdas compression algorithm spectral and ranking documents bibliographic notes exercises matrix is factorization into product three matrices udvt where columns u v are orthonormal d diagonal with positive real entries useful in many tasks here we mention some examples first data close low it nd which good approximation will show that from can get b approximates fact do this every also dened all rectangular or square unlike more commonly used linear algebra reader familiar eigenvectors eigenvalues not assume familiarity realize need conditions on ensure orthogonality contrast called right always form orthogonal set no assumptions left they simple consequence invertible inverse ut as verify gain insight treat rows n points dimensional space consider ...